> [!summary] Column space is the linear transformation of a set of vectors from the domain to a vector in the codomain. > **Key Equations:** $\mathrm{Col}(A)=\mathrm{Span}\left( \left\{ \vec{a_{1}},\dots, \vec{a_{n}} \right\} \right)$ >[!info]+ Read Time **⏱ 2 mins** # Definition The column space of a [[Linear Mapping (Transformations)|linear transformation]] $L: \mathbb{R}^n\to \mathbb{R}^m$ is the [[Sets|set of vectors]] that are [[Linear Mapping (Transformations)|mapped]] onto a vector $\vec{b}\in \mathbb{R}^m$. In other words, this is describing a [[Matrix Notation|matrix]] $A$ whose column space is the [[Span|span]] of its columns. Meaning it contains all possible solutions to $A \vec{x}= \vec{b}$ for some value of $\vec{x}\in \mathbb{R}^n$. Since the result is the [[Span|span]] in $\mathbb{R}^m$, the column space is a [[Subspace|subspace]] of $\mathbb{R}^m$. ## Derivation > [!warning] Assumptions To derive an expression for the column space, assume the following: > - A [[Linear Mapping (Transformations)|linear transformation]] is from $\mathbb{R}^n\to \mathbb{R}^m$ > - The [[Matrix Mapping (Transformations)|matrix mapping]] is defined as $A \vec{x}=\vec{b}$ > - [[Matrix-Vector Multiplication|Matrix vector multiplication]] can be written as $A \vec{x}= x_{1} \vec{a_{1}}+\dots+ x_{n} \vec{a_{n}}$ > - [[Span|Span]] of some [[Linear Combinations|linear combinations]] is written as $\text{Span}(\vec{v_{1}},\dots, \vec{v_{n}})$ $ \begin{align*} \text{Col}(A) &= \left\{ \vec{b} \space | \space \vec{b} = A \vec{x} \in \mathbb{R}^n \right\} \\ &= \left\{ \vec{b} \space |\space \vec{b} = x_{1} \vec{a_{1}}+\dots+x_{n}\vec{a_{n}} \space x\in \mathbb{R}^n \right\} \\ &= \text{Span}(\left\{\vec{a_{1}},\dots,\vec{a_{n}} \right\} ) \end{align*} $ ## Proof of Subspace > [!warning] Assumptions To prove column space is a [[Subspace|subspace]] through the [[Subspace|subspace test]], assume the following: > - The [[Zero Vector|zero vector]] is defined as $\vec{0}_{n}\in \mathbb{R}^n$ and $\vec{0}_{m}\in \mathbb{R}^m$ > - $\exists \vec{u},\vec{v}\in \mathbb{R}^n$ so that $y_{1}=A\vec{u}$ and $y_{2}= A \vec{v}$ > - $c\in \mathbb{F}$ **Zero vector:** $ A\mathbf{0}_n = \mathbf{0}_m \in \operatorname{Col}(A) . $ **Closed under addition:**   $ \mathbf{y}_1 + \mathbf{y}_2  = A\mathbf{u} + A\mathbf{v}  = A(\mathbf{u} + \mathbf{v}) \in \operatorname{Col}(A) . $ **Closed under scalar multiplication:**   $ c \mathbf{y} = c A\mathbf{u} = A(c \mathbf{u}) \in \operatorname{Col}(A) . $ # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/YQRioQ1XUck?si=48MS6nRgxx-0VBdK" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>